A tennis ball dipped in red paint rolls around on the coordinate plane, so that it is at
\((x,y) = (3t^2 - 9t - 5, t^2 - 3t + 2) \)
at time t, where \(0 \le t \le 4 \). Find the length of the paint trail left by the tennis ball.
That's not correct is it Alan ? The function is multi (double ) valued.
The motion between t = 0 and t = 3/2 is retraced between t = 3/2 and t = 3,
and these will simply cancel out if the integral is taken between t = 0 and t = 4, (or between t = 0 and t = 3).
To see what's going on, you have to integrate between 0 and 3/2, between 3/2 and 3 and between 3 and 4.